Subgroups — #13050859

Order 1344.

132 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₃⋊(C₇⋊C₈)	C₂	1	1	yes	maximal, characteristic
2	C₄×C₇×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₃×C₄×C₇⋊C₈	C₂	1	1	yes	maximal, characteristic
4	#89249	—	3	1	no	maximal, Hall
5	C₃×C₄²×C₇	C₂²	1	1	yes	characteristic
6	C₂×C₃⋊(C₇⋊C₈)	C₄	1	2	yes	—
7	C₂×C₇×C₃⋊C₈	C₄	1	2	yes	—
8	C₂×C₃×C₇⋊C₈	C₄	1	2	yes	—
9	C₄×C₇×C₈	—	3	1	no	—
10	C₄×C₇⋊C₈	—	3	1	no	—
11	C₄×C₇⋊C₈	D₃	1	1	yes	characteristic
12	#4455	—	7	1	no	maximal, Hall
13	C₂×C₃×C₄×C₇	C₂×C₄	1	1	yes	characteristic
14	C₂×C₃×C₄×C₇	C₂×C₄	1	1	yes	characteristic
15	C₂×C₃×C₄×C₇	C₂×C₄	1	1	yes	characteristic
16	C₃⋊(C₇⋊C₈)	—	2	2	no	—
17	C₃×C₇⋊C₈	C₈	1	4	yes	complemented
18	C₇×C₃⋊C₈	—	2	2	no	—
19	C₂×C₇×C₈	—	3	2	no	—
20	C₄²×C₇	C₂×D₃	1	1	yes	characteristic
21	C₂×C₇⋊C₈	—	3	2	no	—
22	C₂×C₇⋊C₈	Dic₃	1	2	yes	—
23	C₃×C₄×C₈	—	7	1	no	—
24	C₄×C₃⋊C₈	—	7	1	no	—
25	C₄×C₃⋊C₈	D₇	1	1	yes	characteristic
26	C₂²×C₃×C₇	C₄²	1	1	yes	characteristic
27	C₃×C₄×C₇	C₂×C₈	1	2	yes	—
28	C₃×C₄×C₇	M₁₆	1	2	yes	—
29	C₃×C₄×C₇	M₁₆	1	2	yes	—
30	#322	—	21	1	no	Sylow
31	C₂×C₄×C₇	C₄×D₃	1	1	yes	characteristic
32	C₂×C₄×C₇	C₄×D₃	1	1	yes	characteristic
33	C₂×C₄×C₇	C₂×Dic₃	1	1	yes	characteristic
34	C₇×C₈	—	6	2	no	—
35	C₇⋊C₈	—	6	2	no	—
36	C₇⋊C₈	C₃⋊C₈	1	4	yes	complemented
37	C₂×C₃×C₈	—	7	2	no	—
38	C₃×C₄²	C₂×D₇	1	1	yes	characteristic
39	C₂×C₃⋊C₈	—	7	2	no	—
40	C₂×C₃⋊C₈	Dic₇	1	2	yes	—
41	C₂×C₃×C₇	C₄×C₈	1	1	yes	characteristic
42	C₂×C₃×C₇	#97	1	1	yes	characteristic
43	C₂×C₃×C₇	#97	1	1	yes	characteristic
44	C₄×C₈	—	3	1	no	—
45	C₄×C₈	—	7	1	no	—
46	C₄×C₈	—	21	1	no	—
47	C₂²×C₇	C₄×Dic₃	1	1	yes	characteristic
48	C₄×C₇	ΓL(1,𝔽₂₅)	1	2	yes	—
49	C₄×C₇	ΓL(1,𝔽₂₅)	1	2	yes	—
50	C₄×C₇	C₂×C₃⋊C₈	1	2	yes	—
51	C₂×C₃×C₄	C₄×D₇	1	1	yes	characteristic
52	C₂×C₃×C₄	C₄×D₇	1	1	yes	characteristic
53	C₂×C₃×C₄	C₂×Dic₇	1	1	yes	characteristic
54	C₃×C₈	—	7	4	no	—
55	C₃⋊C₈	—	2	2	no	—
56	C₃⋊C₈	—	14	2	no	—
57	C₃×C₇	#322	1	1	yes	characteristic, complemented, Hall
58	C₂×C₈	—	3	2	no	—
59	C₂×C₈	—	7	2	no	—
60	C₂×C₈	—	21	2	no	—
61	C₄²	D₃×D₇	1	1	yes	characteristic
62	C₂×C₇	C₄×C₃⋊C₈	1	1	yes	characteristic
63	C₂×C₇	#815	1	1	yes	characteristic
64	C₂×C₇	#815	1	1	yes	characteristic
65	C₂²×C₃	C₄×Dic₇	1	1	yes	characteristic
66	C₃×C₄	C₈×D₇	1	2	yes	—
67	C₃×C₄	#1131	1	2	yes	—
68	C₃×C₄	#1136	1	2	yes	—
69	C₂×C₄	Dic₃×D₇	1	1	yes	characteristic
70	C₂×C₄	D₃×Dic₇	1	1	yes	characteristic
71	C₂×C₄	D₂₁⋊C₄	1	1	yes	characteristic
72	C₈	—	6	2	no	—
73	C₈	—	7	4	no	—
74	C₈	—	42	2	no	—
75	C₇	#4455	1	1	yes	characteristic, complemented, Sylow
76	C₂×C₃	#6377	1	1	yes	characteristic
77	C₂×C₃	C₈×Dic₇	1	1	yes	characteristic
78	C₂×C₃	#6389	1	1	yes	characteristic
79	C₂²	Dic₃×Dic₇	1	1	yes	characteristic
80	C₄	D₇×C₃⋊C₈	1	2	yes	—
81	C₄	#66558	1	2	yes	—
82	C₄	#66559	1	2	yes	—
83	C₃	#89249	1	1	yes	characteristic, complemented, Sylow
84	C₂	C₃⋊C₈×Dic₇	1	1	yes	characteristic
85	C₂	#10620488	1	1	yes	characteristic
86	C₂	#10620489	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, hypercenter	C₄²	D₃×D₇
commutator	C₂×C₃×C₇	C₄×C₈
Fitting, cosocle	C₃×C₄²×C₇	C₂²
socle, prime radical	C₂²×C₃×C₇	C₄²
hypocenter	C₃×C₇	#322
LCS	G ▷ C₂×C₃×C₇ ▷ C₃×C₇
UCS	C₁ ◁ C₄²

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
17. C₃×C₇⋊C₈	C₈	semidirect
36. C₇⋊C₈	C₃⋊C₈	semidirect
57. C₃×C₇	#322	semidirect
75. C₇	#4455	semidirect
83. C₃	#89249	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₄×C₃⋊(C₇⋊C₈)	2
C₄×C₇×C₃⋊C₈	2
C₃×C₄×C₇⋊C₈	2
#89249	3
#4455	7